As for the Higgs portal coupling, we consider
\(\lambda _{3}\in [0.001,1]\)  for the WIMP scalar scenario and \(\lambda _{3}\in [10^{-13},10^{-10}]\)  for the FIMP scalar scenario. During the scan, we require the relic density satisfies the Planck observed result in the \(3\sigma \)  range [1]}, i.e., \(\Omega _\chi h^2\in [0.117,0.123]\) . To obtain the specific scenario \(m_N>m_\phi +m_\chi \)  and \(m_\phi >m_\chi \) , these conditions are also imposed manually.
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